# Why do we typically only teach high-school students affine transformations of elementary functions?

A standard pre-calculus curriculum consists of the study of elementary functions: Polynomials, rational functions, (circular and hyperbolic) trigonometric functions, exponential functions, their inverses, and sums, products and composites of these. Moreover, when it comes to understanding transformations, we teach only affine transformations: dilations, reflections (about the axes, or $$y=x$$), and translations.

I originally answered this question to myself as: Well, affine transformations preserve the elementary functions. For example, "all" sine functions are of the form $$f(x) = a \sin(bx + c) +d,$$ which means that all sine functions are described by affine transformations of $$x$$ and affine transformations of $$\sin(x)$$.

But since we allow composites in the definition of elementary functions (or, at least wikipedia does), my previous answer has no meaning.

This leads me to ask the following (potentially meaningless and uninteresting question): Is there a mathematically interesting answer to why affine transformations are distinguished when studying graphical transformations?

Note: While the question is motivated by pedagogy, I'm not interested in "affine is the easiest, so that's why we teach it" -- This answer is unenlightening and no fun.

Edit: In the comments, Matt F. asked Are the affine transformations the largest possible Lie group of (everywhere-defined) transformations of the plane? While this is not my question, it is this flavour of question that I am asking.

• I do find the question to be meaningless. Why study Euclidean Geometry (properties of shapes invariant under isometries or similarities) instead of affine geometry, inversive geometry, hyperbolic geometry, neutral geometry, etc? The answer is cultural: there is no "mathematical reason" to prefer one structure to another. Mar 24 at 12:15
• Could you give a few examples of the type of non-affine transformations you think would be valuable to include in the high school curriculum? Mar 24 at 13:21
• @user2913 We only teach this subgroup because graphs of functions are transformed into graphs of functions under these transformations. Other affine transformations could transform functions into relations which are not functions (ex: rotate the squaring function 90 degrees clockwise). Mar 24 at 14:14
• You claim your question is motivated by pedagogy, but I sure don't see how that is so. Can you explain that? Mar 24 at 16:41
• @user2913 Old school analytic geometry books included rotations of arbitrary relations. I think it has been dropped from curricula because of difficulty. We certainly do use non-affine transformations sometimes, as in a "log-log" plot. Mar 24 at 16:48

We can think of these transformations from two perspectives: as mappings of the plane or as graphs of pre/post composition of the function.

For instance, the map $$T: \mathbb{R}^2 \to \mathbb{R}^2$$ given by $$(a,b) \mapsto (a+2,b)$$ is the translation of the plane two units to the right.

Let $$\phi: \mathbb{R} \to \mathbb{R}$$ be the function $$\phi(x) = x-2$$.

Let $$f:\mathbb{R} \to \mathbb{R}$$ be an arbitrary function, and let $$\Gamma_f$$ be the graph of this function.

Then it is a theorem that $$T$$ maps $$\Gamma_f$$ bijectively onto $$\Gamma_{f \circ \phi}$$. This theorem is usually stated in much less awesome language in precalculus textbooks.

In precalculus, we almost entirely restrict ourself to the second perspective of function pre/post composition. The first notion is almost entirely absent: at this level, we only treat real valued functions "of one variable", so mappings like $$T$$ are not defined or even thought of as functions.

In precalculus we do think about pre/post composition with functions other than affine functions. This happens every time we graph a composition of two functions like $$\sin(x^2)$$. It also happens with "log/log" plots. We just do not generally think of it as a "transformation" of the graph, because we do not have a notion of a mapping of the plane to itself. Even with the affine functions this is only done informally.

Pre/post composition with affine functions is an important special case with important properties. In particular, we can reason about these using the theorems and techniques of high school geometry (while we cannot similarly reason about more general mappings of the plane, requiring either linear algebra for more general affine mappings, or multivariable calculus for smooth mappings).

We do not generally consider other affine transformations because (as noted) we do not consider functions $$\mathbb{R}^2 \to \mathbb{R}^2$$ at all, if we did they do not correspond to function pre/post composition, they do not take graphs of functions to graphs of functions, and they require too much linear algebra to be understandable to students at this level. The one exception (as you note) is reflection in $$y=x$$, which is important because it corresponds to taking the inverse of a function.

Moreover there are extremely practical reasons to be concerned with pre/post composition with affine functions: they represent "affine coordinate" changes, which come up all the time in real life!

If $$F(t)$$ represents the temperature in Fahrenheit $$t$$ hours after noon, and $$C(t)$$ represents the temperature in Celsius $$t$$ minutes after 8:00am, then

$$F(t) = \frac{9}{5}C(60 \cdot (t+4)) + 32$$

or in other words

$$F = \phi_2 \circ C \circ \phi_1$$

Where $$\phi_1(t) = 60 \cdot (t+4)$$ is the conversion from "hours after noon" to "minutes after 8:00am" and $$\phi_2(x) = \frac{9}{5}x+32$$ is the conversion from Celsius to Fahrenheit.

Since affine changes of coordinates are so ubiquitous in the universe (and are indeed the basic building blocks of all other coordinate changes: this is called linear algebra and multivariable differential calculus), it makes sense to attend to the basics in high school!

• I sometimes use the Celsius to Fahrenheit formula as a good example for why swapping the variables to calculate $f^{-1}(x)$ is a bad practice. We gain much by using separate letters for the domain and range in applied problems. Mar 25 at 20:37

I'll give a less elegant answer:

because the dilations and translations are easy to understand and implement.

If we were interested in preserving shape under the transformations then surely rigid motions or orthogonal transformations would be a better scope for the discussion. Sadly, highschool students (in the USA) do not typically learn linear algebra, much less use of eigencooordinates to orthonormally diagonalize quadratic forms. This material is not particularly difficult, for example Lay's Linear Algebra text is fairly elementary and includes a section. Some of the better Calculus texts also include rotated forms of conic sections (Anton does at least in the version I looked at last)

If I wanted to make the graphical transformations interesting to the type of graph considered I might try asking:

• what class of transformations maintains the graph type for a given type of graph

For example, which class takes lines to lines, or parabolas to parabolas, or rational functions to rational functions. There is a lot of flexibility in these questions and I suspect they would be rather challenging to curious highschool students (which exist).

I think there's a lot of variation in content of precalculus. I had a semester of theory of functions and semester of analytic geometry, both including some aspects of calculus itself, in a strong public school. But in some ways you could have skipped that stuff and moved right into calculus after algebra two trig.

Rotations were definitely included. And looking in analytics books, are still a normal topic. Didn't kill me to have them, but wouldn't have missed them either. Not a technique I remember using much in standard stem undergrad courses. And lot of long formulas to write and remember. But arguably easier to look up now and use than if you looked them up for first time with no working experience.

But big picture I would be wary of, one, assuming your experience is typical or should be. And two, there are always competing priorities of time and difficulty versus content.

P.s. I find the term affine transformation a little theoretical. Not sure if that's how discussed by normal students or teachers of precalc. But has been a long time so apologies if that is how topic is discussed in high school.